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<h1>Physics 137A: Quantum Mechanics</h1>
<h2>Wednesday, January 18</h2>
<p>Wick Haxton
haxton@berkeley.edu
431 Old LeConte</p>
<p>GSI: Grant Larsen
420B Old LeConte
larsen.ge@berkeley.edu</p>
<p>Decide office hours during discussion next week.</p>
<h1>Generic Introduction</h1>
<p>special relativity; new theory returns to old theory as some ε→0.
ℏ = h/2π (Planck's constant – angular momentum).</p>
<p>(E₁-E₂)̱\bar{E} = (E₁-E₂)/(p²/2m)&gt;(πℏ/pd)² ≈10^{-69}
ℏ = h/2π = 1.05·10^{-34} J·s
Physics 137A: Quantum Mechanics
===============================
Friday, January 20</p>
<hr />
<h1>Historical review</h1>
<p>Planck: energy is infinite. (remove this paradox by quantization
of all matter.)</p>
<p>T{blackbody} &gt; T{environment} → radiates</p>
<p>encountered contradiction by looking at higher and higher energy modes
in the box.</p>
<p>Incorrect implicit assumption was that energy is not continuous
(i.e. arrives in discrete "packets").</p>
<p>(dimensional analysis)</p>
<p>[ Planck radiation formula. ]</p>
<p>[ photoelectric effect also consistent with quantization of light. ]</p>
<p>de Broglie: particles behaving as waves, in addition to waves behaving as
particles.</p>
<p>(double-slit experiment)</p>
<p>"NOT an effect due to the interference of different photons passing
through the two slits. lower the light intensity so that on average
only one photon is in flight at one time. The intensity pattern is
still produced once the experiment is carried out long enough to
accumulate counts → the interference effect is a property of a single
photon. BUT pattern disappears if one slit is closed."</p>
<p>(superposition of multiple states.)</p>
<p>(Schrödinger's wave equation. blah. Dirac's relativistic version)</p>
<p>(uncertainty &amp; conjugate bases)</p>
<p>(fourier analyses, more about uncertainty)</p>
<h2>Schrödinger's wave equation</h2>
<p>iℏ(∂ψ(x,t)/∂t) = [-(ℏ²/2m)(∂²/∂x² + V(x)] ψ(x,t) = H(x)ψ(x,t)
[ only one-dimensional, so far ]</p>
<p>Operators now written with a circumflex.
Physics 137A: Quantum Mechanics
===============================
Monday, January 23</p>
<hr />
<p>Use of Mathematica/Matlab/WolframAlpha encouraged for homework.</p>
<p>Recap. Talk about <em>how</em> uncertain, measured by product of two related
properties, e.g. momentum + position Schrödinger's equation.</p>
<p>Classical mechanics teaches us (incorrectly!) that physics is
deterministic. This is actually just the most likely path that you can
follow from point a to point b.</p>
<p>The issue is that we have Heisenberg's uncertainty principle, so it is
impossible to precisely know momentum and position simultaneously.</p>
<p>three primary interpretations of QM
 * Einstein - deterministic (hidden variables). Wrong.
   + "God does not play dice."
   + Not enough information to fully ascertain position of particle;
     must be an issue with the theory.
 * Copenhagen (Bohr) - nondeterministic. Widely accepted.
   + "orthodox position", 
   + Actually does exist in all possible results simultaneously until
     measurement is made and wavefunction is collapsed.
 * "agnostic position": who cares?</p>
<p>variance is 〈(Δi)²〉 = 〈i²〉 - 〈i〉²</p>
<h1>Physics 137A: Quantum Mechanics</h1>
<h2>Wednesday, January 25</h2>
<p>Discrete distributions and somesuch.</p>
<p>moments and stuff. mean: first moment. variance: roughly second
moment. skewness: third moment. kurtosis (wtf?): fourth moment.</p>
<p>Moments are the analogue of derivatives for distributions. Sort
of. You also have derivatives, which aren't very useful, usually.</p>
<p>stuff with normalizing wave function, |ψ|² = ψ*ψ, etc.
Physics 137A: Quantum Mechanics
===============================
Friday, January 27</p>
<hr />
<p>Talk about using WolframAlpha to compute integrals.</p>
<p>blah, complex numbers. |Ψ|² ≡ ΨΨ*</p>
<p>Satisfying Schrödinger's equation in itself guarantees that
normalizing at a fixed point in time guarantees normalization for all
time.</p>
<p>[ derivations, consulting of notes. Correcting of errors. ]</p>
<p>[ working out expectation of momentum from the wave function. ]</p>
<p>〈v〉 = (d/dt)〈x〉 = [ℏ/im ∂/∂x]{Ψ}Ψ*
m〈v〉 = 〈p〉
\hat{p} = [ℏ/i ∂/∂x]</p>
<p>〈Ô〉 ≡ ∫[Ô]{Ψ}Ψ*dx</p>
<p>ΔxΔp ≥ ℏ/2
e^{ip·x}</p>
<p>Fourier transform of the Gaussian:
P(x) = 1/(√(2π)σ) e^{-x²/(2σ²)} = 1/(2π) ∫[cos(px/ℏ) e^{-p²σ²/(2ℏ²)}]∂(p/h)
Physics 137A: Quantum Mechanics
===============================
Monday, January 30</p>
<hr />
<p>Nothing that hasn't already been covered in statistical mechanics.</p>
<p>fourier transform blah blah blah</p>
<p>quaternions and stuff</p>
<p>Stationary states ⇔ solving the Schrödinger equation (time independent).</p>
<p>iℏ ∂Ψ/∂t = (-ℏ²/2m ∂²/∂² + V(x))Ψ
         = (T + V)Ψ
V = V(x,t) = V(x)
Ψ(x,t) = Ψ(x)e^{-iEt/ℏ}</p>
<p>Eliminated time dependence with Fourier transform.</p>
<p>Time independent Schrodinger equation:
EΨ′(x) = [-ℏ²/2m ∂²/∂x² + V(x)]Ψ′(x)
Ψ(x,t) = Ψ′(x) e^{-iEt/ℏ}</p>
<p>Probability density is fixed over time. Very strong statement.</p>
<p>Such a solution is called a stationary state because the probability
density is independent of time. i.e. V(x,t) → V(x)</p>
<p>First big idea, stationary states.
〈Ĥ〉 = ∫Ψ⁺ĤΨdx</p>
<p>σ{Ĥ} = 〈H²〉 - 〈H〉² = 0. Stationary states have definite
energy. Eigenvalues.
Physics 137A: Quantum Mechanics
===============================
Wednesday, February 1</p>
<hr />
<p>REMINDER: surfaces ⇒ boundaries.</p>
<p>Mostly duplicated information.</p>
<p>variance in energy is 〈H²〉 - 〈H〉² = 0.</p>
<p>proof of orthogonality; partial integrations and stuff. For two
different eigenvectors.</p>
<p>〈2|\hat{O}|1〉.</p>
<p>If you combine results with other results, we have a Krönecker delta.
∫Ψ*{i}Ψ{j} = δ{ij}.</p>
<p>The basic idea is that if you just focus on the time-independent
Schrödinger equation, it's quite powerful and used over and over again
in quantum mechanics.</p>
<ul>
<li>First idea is time-independent solutions of the Schrödinger equation.</li>
<li>Basis vectors don't change over time.</li>
<li>Second idea is stationary states or eigenvalues or whatever (definite
   energy).</li>
<li>Basis vectors are eigenvectors.</li>
<li>Third idea is orthogonality.</li>
<li>These basis vectors are orthonormal.</li>
<li>Fourth idea is completeness.</li>
<li>This basis is a complete basis.</li>
</ul>
<p>parity? arbitrary coordinates. V(x) is infinite outside the well, 0
inside the boundaries. We want to know how to solve that equation with
quantum mechanics. blah blah blah, construct Fourier series
already. Nothing interesting.</p>
<p>Ψ(x) = Asin(kx) + Bcos(kx).</p>
<p>We want our divergence to be 0. Continuous, zero at boundaries. Must
be zero at boundaries. THESE ARE BASIC BOUNDARY CONDITIONS.
½ka = π/2 + ℓπ  (ℓ ∈ \mathbb{N})
  a = (2ℓ+1)π/k</p>
<p>blah, constructing fourier series still
Physics 137A: Quantum Mechanics
===============================
Friday, February 3</p>
<hr />
<p>Observations:</p>
<ul>
<li>
<p>Eigenvalues discrete</p>
<p>⇔ imposed by boundary conditions
 Ψ(x) = 0 at x = a/2, -a/2.</p>
</li>
<li>
<p>wavefunction with En ∝ n²</p>
</li>
</ul>
<h1>Properties</h1>
<ul>
<li>Even/odd states under reflection around origin</li>
<li>
<p>V(x) = V(-x).</p>
</li>
<li>
<p>Wave functions differ by the number of nodes.</p>
</li>
<li>
<p>n - 1 = # nodes.</p>
</li>
<li>
<p>Wave functions are orthonormal. 〈Ψ{m}|Ψ{n}〉</p>
</li>
<li>Wave functions are also complete.</li>
</ul>
<p>blarg, constructing fourier series (finally) and bra-ket notation
(only now?).
Physics 137A: Quantum Mechanics
===============================
Monday, February 6</p>
<hr />
<p>Anything prepared in an eigenstate stays in an eigenstate. But what if we
don't prepare our state in this way?</p>
<p>You can expand this wave function in terms of this complete basis, and you
have to write down the initial equations that you got. Guess. That guess is
right. iℏ∂/∂t Φ = [-ℏ²/2m ∂²/∂x² + V] Φ</p>
<p>iℏ ∑a{n}Ψ{n}(-iEn/ℏ)exp(-iEnt/ℏ) = ∑[a{n}exp(-iEnt/ℏ)[-ℏ²/2m d²/dx² + V(x)]Ψ{n}]</p>
<h1>Summary</h1>
<h1>V(x,t) = V(x)</h1>
<h1>We solve for special stationary states ⇔ [-ℏ²/2m d²/dx² + V]Ψ = EΨ ⇒</h1>
<p>complete set {Ψ, E} n ∈ N
  |Ψ|² = |Ψ|² end of time</p>
<h1>General solution</h1>
<p>Φ = ∑aΨ</p>
<p>〈Φ|H|Φ〉 = ∑|a{n}|²E{n}</p>
<p>stuff with conjugates and whatnot.
Physics 137A: Quantum Mechanics
===============================
Wednesday, February 8</p>
<hr />
<p>Harmonic oscillator</p>
<pre><code>V(x)| __          ____
    |/  \        /
    |    |   ___/
    |    \__/
    |_________________
                     x
</code></pre>
<p>This potential is symmetric in momentum and position. A wave packet in
coordinate space, if you Fourier-transform it, is the same wave packet is
momentum space. Has this property that if you want to do many-body physics
(not in this class) and use as a basis the harmonic oscillator, then you
get nice stuff that happens. One of the great things about the harmonic
oscillator is that, say you have 3 particles. Three coordinates describing
them. Most of the time, you're not interested in said coordinates, but
rather what the wave function looks like. Take out center-of-mass
motion. You have the same problem in general many-body theory. Excitations
in nucleus, whatever. Would love to have a big set of basis states where
you could write things down simply and just work with the interesting
part. The harmonic oscillator allows you to do that. For all these reasons
this problem is actually important, but you don't see them in the
textbook. tl;dr: this makes life easier when you get to harder problems.</p>
<p>You recognize such a potential because of springs. W = kx²/2. Because the
potential is quadratic, the restoring force is linear. You can calculate
the frequency omega ≡ √(k/m)</p>
<p>Let's start by writing down the time-independent Schrödinger
equation. There's a potential term, and that we'll write as ½momega². What prof
is going to do is straight out of the book, so don't take notes. Cool.</p>
<p><strong>Adding 0 and multiplying by 1: most useful math tricks.</strong></p>
<p><strong>Addition/subtraction: physicist's worst nightmare.</strong></p>
<p><strong>Regularization: it is your best friend.</strong>
Physics 137A: Quantum Mechanics
===============================
Friday, February 10</p>
<hr />
<p>So, last time we were doing a harmonic oscillator. Calculation takes two
full lectures. Energy has to be quantized. Good first start. Lots of
motivation for the harmonic oscillator. Useful in general for lots of
different interactions. The first step is to define the time-independent
solutions. First thing to do is to introduce a dimensionless coordinate.</p>
<p>Talk about solution to simple harmonic oscillator. Why it must be a finite
polynomial (actually, no good reason -- it's just easier). how to get said
finite polynomial. numerator has to vanish. result must be either odd or
even. it's in the math. working out hermite polynomials by hand. haven't
even gotten around to the closed-form version.</p>
<p>(n+1)(n+2)T(n+2) = (2n+1 - 2E/ℏomega)T(n)</p>
<p>δ(nm)√π 2ⁿn!. Orthogonal.</p>
<p>Recurrence relation. H{n+1} = 2yH{n} - 2n(H{n-1}.</p>
<p>dⁿH{n}/dn = 2nH{n-1}.</p>
<p>Now you just have to go ahead and plug in this formula into our wave
function. It's just a whole lot of algebra, so Haxton will just write down
the result instead of possibly getting it wrong. Will probably be in the
text.</p>
<p>You have to choose either a₀ or a₁ to be 0. Choosing both yields a useless
result.</p>
<p>blarg, orthog basis saves time.</p>
<p><strong>Protip: You can violate energy conservation as long as you don't do it
too much.</strong>
|x| &gt; √(ℏ/momega (2n+1)) ⇒ ∫Ψ*x²Ψdx.</p>
<p>whenever you have an x that is more than twice its root-mean-square, you
are violating mathematics. Which is great. This happens 11% of the
time. This is excellent.</p>
<p>Next Monday: calculating expectation value of p².
Physics 137A: Quantum Mechanics
===============================
Monday, February 13</p>
<hr />
<p>Hermite relations:
∫Hm(y)Hn(y)e⁻{y²/₂}dy = √π 2ⁿ n! δ{nm}
H{n+1} = 2yH{n} - 2nH{n-1}
dH{n}/dy = 2nH{n-1}.</p>
<p>〈x²〉n = ∫Ψ*(x)x xΨ(x)dx. Use recurrence relation, then use first
relation.</p>
<p>〈p²〉 = momegaℏ²(2n+1)/2. 〈T〉 = 〈p²〉/2m = ℏω/2 (2n+1)/2</p>
<p>σ{x}σ{p} = ℏ/2 (2n+1) ≥ ℏ/2 (n guaranteed to be non-negative)</p>
<p>Barrier presentation. Potential greater than energy.</p>
<p>Talk about free particles.
Physics 137A: Quantum Mechanics
===============================
Wednesday, February 15</p>
<hr />
<p>Recap: free particle schrödinger equation has a set of solutions. \Psi(x) =
1/√(2π)e^{ikx}. |Ψ(x)|² = 1/2π. Not normalizable. Analogous to sin, cos
functions in a box. -∞ &lt; k &lt; ∞. <em>both signs</em>. E{k} = ℏ²k²/2m</p>
<p>Finite a box. from -πa to πa.</p>
<p>e^{ikx} as our wave functions.</p>
<p>Boundary conditions: e^{-ikπa} = e^{ikπa}. PERIOD ≡ 1/a; freq ≡ a by
construction.</p>
<p>We now have a discrete set of wave functions over a finite width. We know
how to normalize this. Do what we would normally do. Our interval is (as
stated earlier) [-π/a,π/a]. blah, demonstration of orthonormality. Take
limit as a → ∞.</p>
<p>talk about boxes. lots of handwaving. sucky explanation of δ(r).</p>
<p>Now that we have our δ (finally!) we can just write down the answer. Or
not.</p>
<p>When all the dust clears, what do you get? What we've discovered from these
arguments is that we wanted to get these a{k}. So we had the right idea,
(not really) and we worked out the math by hand and had to check for
orthonormality as opposed to completeness.</p>
<p>We naturally get a(k′). "If you squint at this, it looks just like our
Krönecker delta."</p>
<p>The final step is that we write the solution. Evolves just as you expected
it to be. Plug it back into the Schrödinger equation.</p>
<p>(blarg, this is the Fourier transform. Finally. The <em>obvious</em> way to do the
problem.)</p>
<p>wave packets, momentum composition. blah blah blah, you can show that
they're equivalent, etc.
Physics 137A: Quantum Mechanics
===============================
Friday, February 17</p>
<hr />
<h1>finite square well???</h1>
<h1>pset -&gt;</h1>
<pre><code>week from Monday.
</code></pre>
<h1>midterm -&gt;</h1>
<pre><code>no  δ function potential.
yes δ-function ⇔ Fourier
up to bound-state square well
</code></pre>
<h1>notes -&gt;</h1>
<pre><code>Saturday
</code></pre>
<h1>schedules/accommodations -&gt; Thursday?</h1>
<p>fourier transform! fourier transform!</p>
<h1>Finite square well (finally)</h1>
<p>We'll set some zero energy, and a potential that has a depth of -V₀. This
is the base of our well. My wells are always symmetric around the origin,
and they always have width a. So we've got two regions: E &gt; 0 and E &lt; 0.</p>
<p>Negative energy: "is bound". It's just a shift of our ground. No physics
involved whatsoever. So we want to solve this problem. We'll do both bound
state and continuum solutions for this thing.</p>
<p>Curvature of the wave function corresponds to energy. If I look at the
energy (relative to 0) of the first and second eigenstates, the energy gaps
are going to be smaller than higher up.</p>
<p>Not an infinite number of energies, since it's not an infinite well. Gets
increasingly compact on the way up.</p>
<p>Continuous stuff.</p>
<p>Let's first consider stuff in our well. We've got two regions: x&lt;-a/2,
x&gt;a/2, and -a/2 &lt; x &lt; a/2. The first part is the free SE. Nothing
interesting there.
Physics 137A: Quantum Mechanics
===============================
Wednesday, February 22</p>
<hr />
<p>No cheatsheet. Won't need it. Mostly qualitative questions. One really hard
question for bored people. Five points. Believe it or not, we're catching
up on the homework. Too much turmoil, so these will just be in the
discussion sections.</p>
<p>So what we were doing last time, we were halfway through the wave
function. Derivatives have to be continuous, functions must be
continuous.</p>
<p>We know is that wave functions either break down into odd solutions or even
solutions. So solve each separately.</p>
<p>Standard solving of ordinary differential equations: write down general
form, then enforce boundary conditions.</p>
<p>\kappa is pre-propagation, so doesn't know about potential. k is
propagation.</p>
<p>Remember that this only gives us half of the solutions.
Physics 137A: Quantum Mechanics
===============================
Monday, February 27</p>
<hr />
<p><mathjax>$\let\onabla\nabla
\renewcommand{\vec}[1]{\overset{\rightharpoonup}{#1}}
\renewcommand{\nabla}{\vec{\onabla}}
\newcommand{\div}{\vec{\onabla}}
\newcommand{\tens}[1]{\overset{\rightharpoonup\hspace-6pt\rightharpoonup}{\mathbf{#1}}}$</mathjax>
Talk about solving transcendental equations by plotting. More room for
bound states as blue line gets larger. Intersections correspond to
eigenstates. If you're deep down in the bottom of an finite well, it should
look like an infinite well. Your probability of penetrating into the
forbidden region is governed by an exponential in <mathjax>$\kappa$</mathjax>.</p>
<p>Zero-energy bound state (parameter <mathjax>$V_0 a^2$</mathjax> very small) corresponds to an
arbitrarily weak potential and arbitrarily wide well.</p>
<p>Similarly, the <mathjax>$\delta$</mathjax> potential corresponds to <mathjax>$a \to 0$</mathjax>, <mathjax>$V_0
\to \infty$</mathjax>.</p>
<p>Talk about how the wave function has continuous zeroth and first
derivatives at boundaries.</p>
<p>Tunnelling makes the sun work.
Physics 137A: Quantum Mechanics
===============================
Wednesday, February 29</p>
<hr />
<h1>Physics 137A: Quantum Mechanics</h1>
<h1>Transmission, Reflection, Tunneling</h1>
<h2>Friday, March 2</h2>
<p>if we include in our finite basis wave numbers k_n = n\pi/b such that n \gg
b/a can resove bump.</p>
<p>idea: use large basis in n, construct stationary states numerically.</p>
<p>Fourier basis complete over interval.</p>
<p>Stationary state satisfies S.E.
Physics 137A: Quantum Mechanics
===============================
Monday, March 5</p>
<hr />
<p>Handing back of exams.</p>
<p>One repeat problem on next midterm.</p>
<h1>Chapter 3</h1>
<p>Departure from this point; beginning of chapter 3.</p>
<p>Last time was a brief introduction. What we did Friday: Up to now:
differential equation to stationary states. But limiting!</p>
<p>Friday: Found them numerically: basis, then truncated: Physics. You have to
match truncation with Physics you want to capture in the problem.</p>
<p>Numerically diagonalized approximate stationary states.</p>
<p>Why? Fun applications; introduced notions of matrix form of <mathjax>$T + V$</mathjax>,
vectors as solutions to time-independent Schr\"odinger equation.</p>
<p>vectors: <mathjax>$\alpha_i\ket{x_i}$</mathjax>; inner products: <mathjax>$\braket{\alpha}{\beta}
\equiv \alpha^\dagger \beta$</mathjax>: linear transformations/operators: <mathjax>$\op{
\alpha}$</mathjax> (must be Hermitian, even). Representable by a matrix.</p>
<p>Our vector spaces are usually infinite. This should not confuse you.</p>
<p>Quantum vector spaces (Hilbert spaces -- this is technically a lie --
Hilbert spaces in this sense do not contain the zero matrix, nor are they
closed under anything -- it's really more of a k-sphere): examples:
infinite square well: <mathjax>$\sqrt{\frac{2}{a}} e^{i\pi nx/a}$</mathjax></p>
<p>So we know some things about these products. The point about QM is that not
all possible functions are allowable. We know we want to be able to
calculate expectation values of things, which we measure. It's a real
number. Spectral theorem.</p>
<p><mathjax>$\braket{\psi_m}{\psi_n} = \delta_{mn}$</mathjax></p>
<p>What do we do when eigenvalues are degenerate?
 a) Curl up into a ball and cry.
 b) Ignore said fact.
 c) Mathematica</p>
<p>orthonormal basis.
Physics 137A: Quantum Mechanics
===============================
Friday, March 9</p>
<hr />
<p>Quaternions:</p>
<p>[1 0]  [-i 0]  [0 -1]  [0 i]
[0 1], [ 0 i], [1  0], [i 0]</p>
<p>1       i       j      k</p>
<p>\sum_k e_k \ket{\psi_k}</p>
<p>[x,p] = xp - px
Physics 137A: Quantum Mechanics
===============================
Monday, March 12</p>
<hr />
<p>Recap: spectral theorem; completeness of eigenfunctions.</p>
<p>Dirac orthonormality (consider Fourier transform)</p>
<p>Now it gets a little more interesting. Suppose we have some wave packet
(the book calls this generalized statistical interpretation). Not this
example, but the next which will be important. Will tell when you can label
with more than one quantum number.</p>
<p>Reason why this is being done is because you can start thinking about
operators abstractly. Fact that you know the general theorems that go with
any operator, etc.</p>
<p>Let's go through and see what he means by this idea of a generalized
statistical interpretation. Consider operator H (hamiltonian). We'll have
some wave packet <mathjax>$\Phi\equiv \alpha_i\ket{\phi_i}$</mathjax>. The expansion of this
is simply <mathjax>$\braKet{\Phi}{H}{\Phi}$</mathjax>.</p>
<p>There will be an eigenspectrum corresponding to all the possible outcomes
of a measurement. You'll wind up getting this completeness relationship,
where the probabilities correspond to the coefficients.</p>
<p>probability of a particle being at a specific point means nothing in
continuous case -- rather, we have to speak of it being found in some
interval.</p>
<p>Let's write out this idea of simultaneous measurements and commuting
operators.</p>
<p>So now let's go ahead and compute that the commutator is zero.</p>
<p>Another way to measure how different these eigenvectors are is to see
whether these matrices commute or not. If they commute, they have a common
set of eigenvectors. Commuting means <mathjax>$XZ - ZX = 0$</mathjax>. So we want to look at <mathjax>$XZ
- ZX$</mathjax> (a commutator, denoted by <mathjax>$[X,Z]$</mathjax>).</p>
<p>So what does this look like between <mathjax>$\hat{x}$</mathjax> and <mathjax>$\hat{p}$</mathjax>? We have product
rule coming into play... yielding <mathjax>$[x,p] \equiv i\hbar$</mathjax>.</p>
<p>We'll use this to derive <mathjax>$\Delta x\Delta p = \hbar/2$</mathjax>.</p>
<p>Recall: given an observable A and a state <mathjax>$\ket{Ψ}$</mathjax>, the expected value is
<mathjax>$\braKet{\Psi}{A}{\Psi}$</mathjax>. We also saw that the variance was <mathjax>$E(x^2) -
E(x)^2$</mathjax>, so in this case <mathjax>$\sigma^2 = \braKet{\Psi}{A^2}{\Psi} -
\braKet{\Psi}{A}{\Psi}^2$</mathjax>.</p>
<p>Wednesday, what we're going to do is try some derivation of what the
uncertainty principle looks like in general.
Physics 137A: Quantum Mechanics
===============================
Wednesday, March 14</p>
<hr />
<p>playing around with uncertainty principle; derivation through (commutator +
anticommutator)/2. Consideration of Cauchy-Schwarz inequality. Schwarz
inequality becomes equality in certain cases.
Physics 137A: Quantum Mechanics
===============================
Wednesday, March 14</p>
<hr />
<p>Uncertainty relation: what is its meaning? How do you measure a particle?
Observations disturb state.</p>
<p>Uncertainty principle is why we exist. Hydrogen atom: proton surrounded by
some electron orbit. THe size of the proton is roughly <mathjax>$10^{-15} m$</mathjax>, or one
fm (fermi, as we come to know it). THe electron sits a distance <mathjax>$a_0$</mathjax> (Bohr
radius) away from the proton, roughly <mathjax>$5 \cdot 10^{-11}$</mathjax>. Pauli exclusion
principle leads to repulsive force. 1931: Ehrenfest postulated. Later,
1967: Freeman Dyson along with collaborator A. Lenard proved that this
causes normal force: matter would be much squishier.</p>
<p>H = T + V. We know <mathjax>$T = \frac{p^2}{2m}$</mathjax>, and V is given by <mathjax>$-\frac{e^2}{r
4\pi\epsilon_0}$</mathjax>. We thus have <mathjax>$\int d^3x \Psi^*\parens{-\frac{\hbar^2}
{2m}\nabla^2 \Psi}$</mathjax>. This is roughly <mathjax>$R$</mathjax>. Then we know the kinetic energy,
which is the sum of all the momenta, <mathjax>$\frac{1}{2m}\avg{p_x^2 + p_y^2 +
p_z^2} = \frac{1}{2m}3\frac{\hbar^2}{4R^2} \approx \frac{3}{8}
\frac{\hbar^2}{mR^2}$</mathjax>. So <mathjax>$\avg{H} \approx \frac{3}{8}\frac{\hbar^2}{mR^2}
- \frac{e^2}{R4\pi\epsilon_0}$</mathjax>. <mathjax>$\pderiv{}{R}\avg{H} = 0$</mathjax>. Therefore <mathjax>$R =
  \frac{3}{4}\frac{\hbar^2}{mc\alpha}$</mathjax>. </p>
<p>Two good examples of generalizing uncertainty in text. Decay,
travel.</p>
<p>There's another uncertainty relation that's common to write down, i.e. the
time-energy uncertainty principle. Time scale for observable quantity:
width of particle inversely proportional to lifetime.</p>
<p>Let's do a more specific example: consider a wave function given by the
following gaussian wave packet: <mathjax>$\phi = \frac{e^{-(x - x_0)^2 / 2a^2}
e^{ip_0x/\hbar}}{(\pi a^2)^{1/4}}$</mathjax>. What is the uncertainty in the position
of the particle?</p>
<p><mathjax>$\sigma_x = \frac{a}{\sqrt{2}}$</mathjax>.</p>
<p>Assume we have an infinite hilbert space. Note that one is represented by
<mathjax>$\infint dx \abs{xXx}$</mathjax>.</p>
<h1>Dirac's bra-ket Notation</h1>
<p>talk about finding eigenvalues and eigenvectors. <mathjax>$\ketbra{i}{j}H_{ij}$</mathjax>.</p>
<p>Note that <mathjax>$\braket{i}{j} = \delta_{ij}$</mathjax></p>
<p>More bra-ket notation. Basis must span Hilbert space. Completeness
relation.
Physics 137A: Quantum Mechanics
===============================
Monday, March 19</p>
<hr />
<p>Recaps. Uncertainty principle at multiple places. Showed that if you show
momentum acting on Gaussian, it's effectively <mathjax>$-\frac{\hbar}{i} \frac{1}{2
\sigma_x^2} \hat{x}$</mathjax>. Showed that inequality became an equality, so our
fudging worked just fine.</p>
<p>Time dependence shows up explicitly in some operators; none that we've
encountered thus far.</p>
<p>Two things to take away: <mathjax>$\pderiv{}{t}\avg{\hat{Q}_H} = \frac{i}{\hbar}
\avg{\comm{\hat{H}}{\hat{Q}_H}} + \avg{\pderiv{Q_H^\prime}{t}}$</mathjax>. We can use
this to derive the generalized uncertainty principle: <mathjax>$\sigma_H^2\sigma_Q^2
\ge \avg{\frac{1}{2i}\comm{H}{\hat{Q}}}^2\implies \sigma_H\sigma_Q \ge
\frac{\hbar}{2}\abs{\pderiv{\avg{\hat{Q}}}{t}}$</mathjax>. Using this for energy and
time, we show that we've also got a time-energy uncertainty principle.</p>
<p>Interesting things that aren't stationary: e.g. 2p in hydrogen. Somewhat
important, since using a laser, one thing you want to do is pump hydrogen
to an excited state.</p>
<p>Talk about lifetimes being <mathjax>$\delta t \approx \frac{1}{\omega}$</mathjax>.</p>
<p>Bandgaps. Slowing down of atoms. Doppler shift.</p>
<p>More dimensions! Toy models.</p>
<p>Spherical symmetry.</p>
<p>Schrodinger equation in three dimensions: just replace <mathjax>$x$</mathjax> with
<mathjax>$\vec{r}$</mathjax>. Thus <mathjax>$\pderiv{}{x} \to \nabla$</mathjax>.</p>
<p>We have a different normalization, now: it's now an integral over ALL
SPACE. Same reasoning: particle must be <em>somewhere</em>. So, as before, we can
take our wave function and write <mathjax>$\Psi(\vec{r},t) = \psi(\vec{r})
e^{-iEt/\hbar}$</mathjax>. Plug this in, and we get the time-dependent Schrodinger
equation.</p>
<p>Solve this equation, and we'll get the stationary states.</p>
<p>Hydrogen atom! Make use of fact that we'll have various conserved
quantities. This is why we talked about commutators and stuff. Angular
momentum and stuff.</p>
<p>One very interesting class of problems: <mathjax>$V(r)$</mathjax> (not <mathjax>$\vec{r}$</mathjax>). For
instance, <mathjax>$V = \frac{\alpha}{r}$</mathjax>.
Physics 137A: Quantum Mechanics
===============================
Wednesday, March 21</p>
<hr />
<p>Recaps. In 3 dimensions, note that k,p,x are now vectors, and instead of
<mathjax>$kx$</mathjax>, we have <mathjax>$\vec{k}\cdot \vec{r}$</mathjax>.</p>
<p>Beautifully similar to what we have in one dimension. For many problems
we'll be interested in, the potential is spherically symmetric.</p>
<p><mathjax>$$
2\braKet{\psi}{x^2}{\psi} = \braKet{\psi_1e^{-iE_1t/\hbar} +
\psi_2e^{-iE_2t/\hbar}}{x^2}{\psi_1e^{-iE_1t/\hbar} +
\psi_2e^{-iE_2t/\hbar}}
\\ = \braKet{\psi_1e^{-iE_1t/\hbar}}{x^2}{\psi_1 e^{-E_1t/\hbar}} + 
\braKet{\psi_1e^{-iE_1t/\hbar}}{x^2}{\psi_2 e^{-E_2t/\hbar}} + 
\\ \braKet{\psi_2e^{-iE_2t/\hbar}}{x^2}{\psi_1 e^{-E_1t/\hbar}} + 
\braKet{\psi_2e^{-iE_2t/\hbar}}{x^2}{\psi_2 e^{-E_2t/\hbar}}
\\ = \braKet{\psi_1}{x^2}{\psi_1} + 
e^{-i(E_2 - E_1)t/\hbar} \braKet{\psi_1}{x^2}{\psi_2} + 
\\ e^{-i(E_1 - E_2)t/\hbar} \braKet{\psi_2}{x^2}{\psi_1 } + 
\braKet{\psi_2}{x^2}{\psi_2}
\\ = \braKet{\psi_1}{x^2}{\psi_1} + \braKet{\psi_2}{x^2}{\psi_2} +
2\cos\parens{\frac{(E_2 - E_1)t}{\hbar}} \braKet{\psi_1}{x^2}{\psi_2}
$$</mathjax></p>
<p>Solving 3-dimensional separation of variables. Angular equation; radial
equation. Legendre polynomials. <mathjax>$\ell(\ell + 1)$</mathjax></p>